4.3.26 \(y'(x)=a+b \sin (y(x))\)

ODE
\[ y'(x)=a+b \sin (y(x)) \] ODE Classification

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica
cpu = 0.230122 (sec), leaf count = 52

\[\left \{\left \{y(x)\to 2 \tan ^{-1}\left (\frac {\sqrt {a^2-b^2} \tan \left (\frac {1}{2} \sqrt {a^2-b^2} \left (c_1+x\right )\right )-b}{a}\right )\right \}\right \}\]

Maple
cpu = 0.026 (sec), leaf count = 55

\[ \left \{ {1 \left (-2\,\arctan \left ({\frac {\tan \left (1/2\,y \relax (x ) \right ) a+b}{\sqrt {{a}^{2}-{b}^{2}}}} \right ) +\sqrt {{a}^{2}-{b}^{2}} \left (x+{\it \_C1} \right ) \right ) {\frac {1}{\sqrt {{a}^{2}-{b}^{2}}}}}=0 \right \} \] Mathematica raw input

DSolve[y'[x] == a + b*Sin[y[x]],y[x],x]

Mathematica raw output

{{y[x] -> 2*ArcTan[(-b + Sqrt[a^2 - b^2]*Tan[(Sqrt[a^2 - b^2]*(x + C[1]))/2])/a]
}}

Maple raw input

dsolve(diff(y(x),x) = a+b*sin(y(x)), y(x),'implicit')

Maple raw output

1/(a^2-b^2)^(1/2)*(-2*arctan((tan(1/2*y(x))*a+b)/(a^2-b^2)^(1/2))+(a^2-b^2)^(1/2
)*(x+_C1)) = 0